# Diagonalizabilty versus spread for uncertainty (discrete)

Gold Member
I have seen two characterizations of the problem in measuring a discrete variable of a state ψ exactly with each of two non-commuting Hermitian operators A and B:
(1) that the product of the standard deviations ( = √(<ψ|A2|ψ>-<ψ|A|ψ>2), & ditto for B) ≥ 1
(2) that one cannot simultaneously diagonalize the matrix representations of A and B
(i.e., if A = UCU and B = VDV, for unitary U and V and diagonal C and D, with denoting the adjoint, then U≠V.
Where is the link between these two?

stevendaryl
Staff Emeritus
I have seen two characterizations of the problem in measuring a discrete variable of a state ψ exactly with each of two non-commuting Hermitian operators A and B:
(1) that the product of the standard deviations ( = √(<ψ|A2|ψ>-<ψ|A|ψ>2), & ditto for B) ≥ 1
(2) that one cannot simultaneously diagonalize the matrix representations of A and B
(i.e., if A = UCU and B = VDV, for unitary U and V and diagonal C and D, with denoting the adjoint, then U≠V.
Where is the link between these two?

As explained here: http://en.wikipedia.org/wiki/Uncert...2.80.93Schr.C3.B6dinger_uncertainty_relations, the link is that

$\sigma_A \sigma_B \geq \frac{1}{2} | \langle [A,B] \rangle|$